Conventional quantification of stochastic signals through time has generally been performed through either direct analysis of a time series in the time domain (e.g., Rescaled-Range Analysis, Detrended Fluctuation Analysis) or through examination of the power spectrum of the time series upon conversion to the frequency domain (e.g., discrete Fourier transform, fast Fourier transform). Many conventional time series analysis methods tend to focus on patterns, trends, periodicities, correlations, or structure within a signal, the way in which values rise and fall over time. However, by focusing only on the behavior of values of the time series and not addressing the underlying mechanisms of how the time series was generated, these methods provide limited short-term insight into the current and future predictability of the time series.
A digital filter is a computational tool that accepts a sequence of numbers as input and returns a new altered sequence of numbers as output. The sequence of numbers, as a digital signal, represents information about a recorded, measured quantity that varies with time (or position) such as, but not limited to, audio, communications, radio, television, voltages, SONAR, RADAR, medical data (e.g., EEG data), economic data (e.g., stock market prices), environmental data (e.g., water level fluctuations), and positional data (e.g., control systems). A digital filter may be used to integrate, differentiate, smooth, predict, restore, or separate a signal. Digital filters may also be used to eliminate noise from a signal or to model the internal dynamics of the system that generated the signal.
Conventional digital filters employ integer order calculus. In the field of digital signal processing (DSP), the design of a digital filter is a delicate balance between performance, precision, accuracy, and efficiency. While digital filters generally exhibit a high level of performance, conventional digital filter design technology is often limited in ability to accurately perform exact frequency modifications on a signal. In many instances, the filtered signal contains mathematical artifacts of the filtering process (e.g., ripples, wide transition band with slow roll-off, etc.), as approximations of the ideal signal which may manifest as a loss of information in the signal.